1. ## Hahn-Banach Theorem

Let M be a subspace of the normed vector space X, let x lie in X, and let d(x,M)=inf{norm)(x+y); y in M}.

Prove the following results:

i) For all elements f in the Banach dual space X' of X such that norm of f on X' ≤ 1, and f/m (f restricted to m) = 0, and, for all elements y in M,

(mod)f(x)
≤ norm(x+y)

ii) By considering a linear functional on the subspace lin ( M
union {x}), or otherwise, use the Hahn-Banach Theorem to prove that

d(x,M)= sup { f(x): f in X', norm f on X'
≤ 1, f/m =0}

and to show that there exists such an element fo for which fo(x)=d(x,M).

iii) Deduce that M is dense in X if and only if the only element f in X' for which f/m=0 is itself 0
.

Many thanks.

2. (1)Since $|f|\leq1$, and f=0 if restricted to M, then
$|f(x)|=|f(x+y)|\leq|x+y|$ for all y in M.
(2)By (1), we have d(x,M) is greater or equal to sup { f(x): f in X', norm f on X' ≤ 1, f/m =0},
By Hahn-Banach Theorem, there is f satisfies the required condition,
thus the conclusion is proved.
(3)if M is dense in X, then for any element x in X, there is sequence in M converge to the element, since f in X' is continous, thus f(x)=0, for any x in X.
and the "only if" part can be easily proved by Arguement combined with Hahn-Banach Theorem which leads to contradiction.